HMMT 2025 T9

Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f\colon \mathbb{Z}\to \mathbb{Z}$ such that for any integer $x$ ,

$f(x + p) = f(x)$ and $p$ divides $f(x + f(x)) - x$ .